However, for families of irregular graphs, the SNF of the Laplacian matrix is not easy to get in general. So in [1]}, the authors of the present paper determined the sandpile group of a family of irregular graphs by using the relations determined by the cycles and cuts of the graph. Using this method, it is easy to show that the sandpile group of a polygon chain is always cyclic, which has been proved earlier in [2]}, [3]}. Here, we shall follow this idea to study the sandpile group of a (twisted) polygon ring. We first show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, and then we give an explicit relation matrix among these edges. So the sandpile group of any (twisted) polygon ring is the direct sum of at most three cyclic groups. Our result can be used for computing sandpile groups of various infinite families of (twisted) polygon rings, by computing the Smith Normal Form of the relation matrix. Examples include many known families, such as generalized wheel graphs, ladder graphs, Möbius ladders, etc.[4]}, [5]}, [6]}.
